I’m going through Stillwell’s Pillars and trying my hand at finding non-Euclidean distance between two complex numbers in the upper half plane not on the positive y-axis, namely $1 + 2i $ and $2 + i$. I understand that the points lie on the non-Euclidean line that is a semicircle with center at $(0,0)$ and with radius $\sqrt5$, and that I need to perform Möbius transformations in reverse to map the points to the y axis, but the computation is proving to be difficult. Have I made a mistake in understanding or are there points in the transformations that can cause particular problems?
Alternatively, is there a general formula I can apply to determine non-Euclidean distance? I’m aware that when points are on the y-axis the formula is $\lvert\ln(\frac{y_1}{y_2})\rvert$ but not sure how that can be generalized to other points. I’d appreciate any help
Non-Euclidean distance in the upper half plane between two complex numbers
geometryhyperbolic-geometrynoneuclidean-geometry
Best Answer
The Möbius transformation $\phi\colon z\mapsto\frac{z-i}{z+i}$ sends the upper half plane (the set of $z$ closer to $i$ than to $-i$) to the unit disk, sending $i$ to $0$, the imaginary axis to the real axis, and the unit circle to the imaginary axis. If you first divide by $\sqrt 5$, then apply $\phi$, then rotate by 90 degrees, and finally apply $\phi^{-1}$, your points will be on the $y$-axis.